Letfbe a twice differentiable function onRwithf” continuous on [0, 1] such that

the intergral of f(x) dx from (0, 1) = 2 x intergral of f(x) dx from (1/4, 3/4).

Prove that there exists anxoin (0, 1) such thatf”(xo) = 0.

Here is what I have so far....

the intergral of f(x) dx from (0, 1) = 2 x intergral of f(x) dx from (1/4, 3/4)

f(c1) (1-0) = 2 f(c2) (3/4 - 1/4)

for c1 belonging to (0, 1) and c2 belonging to (1/4, 3/4)

f(c1) = f(c2)

By Rolle's Theorem, there exists c3 that belongs to (c1, c2) such that f ' (c3)=0

I need another part similar to this so that the derivitive of these which would be f " (xo) = 0.

Any suggestions are welcome.